100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

Relativity in the Global Positioning System 271becomes irrelevant. (The Sagnac effect on moving ground-based receiversmust still be considered.) Gravitational frequency shifts and second-orderDoppler shifts must be taken into account together. The term Φ 0 in Eq. (23)includes the scale correction needed in order to use clocks at rest on theearth’s surface as references. The quadrupole contributes to Φ 0 in the term−GM E J 2 /2a 1 in Eq. (23); there it contributes a fractional rate correctionof −3.76 × 10 −13 . This effect is large. Also, V is the earth’s gravitationalpotential at the satellite. Fortunately, earth’s quadrupole potential falls offvery rapidly with distance and up until very recently its effect on satellitevehicle (SV) clock frequency has been neglected. This will be discussed in alater section; for the present I only note that the effect of earth’s quadrupolepotential on SV clocks is only about one part in 10 14 .Satellite orbits. Let us assume that the satellites move along Keplerianorbits. This is a good approximation for GPS satellites, but poor if thesatellites are at low altitude. This assumption yields relations with which tosimplify Eq. (23). Since the quadrupole (and higher multipole) parts of theearth’s potential are neglected, in Eq. (23) the potential is V = −GM E /r.Then the expressions can be evaluated using what is known about theNewtonian orbital mechanics of the satellites. Denote the satellite’s orbitsemimajor axis by a and eccentricity by e. Then the solution of the orbitalequations is as follows: 11 the distance r from the center of the earth to thesatellite in ECI coordinates isr = a(1 − e 2 )/(1 + e cos f) = a(1 − e cos E) . (24)The semimajor axis of GPS satellites is a = 27, 561.75 m, chosen so thata given satellite will appear in exactly the same place against the celestialsphere twice per day. The angle f, called the true anomaly, is measuredfrom perigee along the orbit to the satellite’s position. The true anomalycan be calculated in terms of the eccentric anomaly E, according to therelationships:cos f = cos E − e1 − e cos E , sin f = √ 1 − e 2 sin E1 − e cos E . (25)To find the eccentric anomaly E, one must solve the transcendental equation√GMEE − e sin E =a 3 (t − t p ), (26)where t p is the coordinate time of perigee passage.In Newtonian mechanics, the gravitational field is conservative and totalenergy is conserved. Using the above equations for the Keplerian orbit, one